Collection of Aerosol Particles by a Conducting Sphere in an ...

COLLECTION OF AEROSOL PARTICLES 305n=n m [301At r = r, the radial components of the external field for all 8 vanish, and we would expectthat an@- = 0 at Y = r, for all 8. But this requirement is automatically satisfied in view ofEq. [26] since the quantity in the first square brackets vanishes at r = r,.Equation [30] can be written asexp(-z)-exp{$[E,q(r-$)cos*-$11n = n, [311exp -BQ4\ - 1( Dal LBoth the numerator and denominator on theright hand side of Eq. [3 11 are now positive.It is easy to see that the concentration is notuniform at r = r, (remember that r,,, is afunction of 0). The concentration y1,, islargest at cos 0 = 0 since the potential is zeroat r = r,. On the other hand, the smallestn,=, occurs at 0 = 0 since the potential maximumis lowest and therefore the negative ofthe potential maximum is largest (see Fig. 2).Even at a constant r near the sphere, the0”FIG. 3. The surfaces of r = r, calculated from Eq.[20] for different values of h. Note that these surfacesare not equipotential surfaces.concentration n is also smallest along 0 = 0,as can be seen from Eq. [3 11. This is consistentwith the physical consideration becausethis is where the particles meet the strongestrepulsive forces.From Eq. [3 I] we can also see that whenEO increases to an extent that the second termin the numerator equals the first term, theconcentration becomes zero. Further increasein EO would cause the concentrationto become negative. The negative value itselfis unimportant; it simply means that undervery strong external field, particles cannotcome close to the hemisphere because of thevery strong repulsive forces. This is also consistentwith physical reasonings.The solution for the region where cos f3> 0 and r > r, is not obtained yet. On theother hand it should not concern us herebecause the particle flux in this region is basicallydrifting along the field lines and isdirected away from the sphere. Thus, insofaras the collection of aerosol particles by thesphere is concerned, we can ignore this region.3. The calculation of collection kernel. Tocalculate the collection kernel, we need toknow the radial concentration gradients atthe surface of the sphere. Thus by taking theradial derivatives of Eqs. [25] and [31] andset r = a, we obtain.Journal of Co/lord and Interface Science, Vol. 94, No. 2, August 1983